Crossover-Aware Multiscale Attention

1. Integration point: use a 1-D dilated CNN or a transformer for long sequences, or a 2-D CNN/ViT for images. Construct local features with windows or dilations \(R_l<\hat\lambda\), and a global feature branch with windows \(R_l\geq\hat\lambda\). Fuse them with the gate \(g(R_l)\) above. The input may be raw coordinates, tokens, or patch embeddings; no quasicrystal-specific feature is required. 2. Pseudocode: for each layer, compute local attention/features on radius \(R_l\); compute a cheap global pooled or sparse-attention feature only when \(R_l\geq\hat\lambda\); calculate \(g_l=\operatorname{sigmoid}(\alpha\log(R_l/\hat\lambda)+b)\); output \(h_l=(1-g_l)h_l^{local}+g_lh_l^{global}\). Train \(\alpha,b\) jointly, optionally regularizing the gate toward the predicted transition. 3. Estimate \(\hat\lambda\) empirically by measuring a local-versus-global distinguishability curve: train a small probe to distinguish real samples from phase-randomized or shuffled surrogates at window sizes \(R\), and define \(\hat\lambda\) as the first \(R\) where probe accuracy exceeds a fixed threshold such as 0.75. The scaling exponents \(\nu=1\) or \(2\) are imported from the paper when generating controlled data with symmetry order \(N\); they are not assumed for arbitrary datasets. 4. First cheap experiment: generate 1-D Fibonacci/quasicrystal sequences with several effective orders or use deterministic sums of incommensurate sinusoids, together with matched shuffled sequences. Compare a standard local CNN, full-attention transformer, and the gated model at equal FLOPs. Sweep context length \(R\), plotting classification accuracy and gate activation. The falsifiable prediction is that local-only accuracy stays close to chance for \(R/\lambda_N<1\), while accuracy rises near \(R/\lambda_N\approx1\); fitting transition locations across \(N\) should give slope \(\nu\approx1\) in 1-D and, for 2-D generated tilings, \(\nu\approx2\) on a log-log plot. A successful implementation should also reduce global-attention computation when \(\hat\lambda\) is large without losing post-crossover accuracy.